The area of a triangle is 2x^2-11x+15. If the triangle's base is 2x-5, what is its height?

Jul 2, 2018

$\text{height} = \textcolor{red}{2} \left(x - 3\right)$

Explanation:

Recall;

Area of triangle $= \frac{1}{2} \text{base" xx "height}$

Area of triangle $= 2 {x}^{2} - 11 x + 15$

base $= 2 x - 5$

Plugging in the given;

$2 {x}^{2} - 11 x + 15 = \frac{1}{2} \left(2 x - 5\right) \times \text{height}$

$2 {x}^{2} - 11 x + 15 = \frac{2 x - 5}{2} \times \text{height}$

$\frac{2 {x}^{2} - 11 x + 15}{1} = \frac{2 x - 5}{2} \times \text{height}$

$2 \left(2 {x}^{2} - 11 x + 15\right) = \left(2 x - 5\right) \times \text{height}$

$4 {x}^{2} - 22 x + 30 = \left(2 x - 5\right) \times \text{height}$

$\frac{4 {x}^{2} - 22 x + 30}{2 x - 5} = \text{height}$

$\left(4 {x}^{2} - 22 x + 30\right)$

Simplifying;

$\frac{4 {x}^{2}}{2} - \frac{22 x}{2} + \frac{30}{2}$

$2 {x}^{2} - 11 x + 15$

Using Factorization Method..

$6 \mathmr{and} 5$ are factors..

$2 {x}^{2} - 6 x - 5 x + 15$

Grouping;

$\left(2 {x}^{2} - 6 x\right) \left(- 5 x + 15\right)$

$2 x \left(x - 3\right) - 5 \left(x - 3\right)$

$\left(x - 3\right) \left(2 x - 5\right)$

Therefore;

$\frac{4 {x}^{2} - 22 x + 30}{2 x - 5} = \text{height}$

$\textcolor{w h i t e}{\times \times x} \downarrow$

$\frac{\textcolor{red}{2} \left(x - 3\right) \left(2 x - 5\right)}{2 x - 5} = \text{height}$

$\frac{\textcolor{red}{2} \left(x - 3\right) \cancel{2 x - 5}}{\cancel{2 x - 5}} = \text{height}$

$\textcolor{red}{2} \left(x - 3\right) = \text{height}$

Jul 2, 2018

color(maroon)("Height of the triangle " h = 2 (x - 3)

Explanation:

$\text{Area of triangle = (1/2) * (base * height)}$

${A}_{t} = \left(\frac{1}{2}\right) b h$

$\text{Given : } {A}_{t} = 2 {x}^{2} - 11 x + 15 , b = \left(2 x - 5\right)$

$h = \left(\frac{2 \cdot {A}_{t}}{b}\right)$

$h = \frac{2 \cdot \left(2 {x}^{2} - 11 x + 15\right)}{2 x - 5}$

$h = \frac{2 \cdot \cancel{2 x - 5} \cdot \left(x - 3\right)}{\cancel{2 x - 5}}$

$h = 2 \cdot \left(x - 3\right)$